Note.—The security effected by selecting a large radius
for the intersections, referred to in last problem, is still
more important in the case before us, as a line may be
truly bisected even though the bisecting line be obliquely
drawn; while the perpendicularity is destroyed.

In this way, a perpendicular may be drawn through
the middle of a board or a sheet of paper, by one operation,
when a base-line is laid down.

In this way, also, radial lines may be drawn to a circular
arc, without reference to the centre, as the line C D plainly
passes towards the centre of the arc A F B.

Problem IV.—To draw a perpendicular to a straight
line, from a given point in that line.

1st Method.—With any radius, from the given point
A, in the given line B C, cut (f-w. 63.)

the line at B and C; with a
greater radius describe arcs
from B and C, cutting each
other at D, and draw the per-
pendicular D A.

2 d Method. — From any
centre F, above B C, describe a circle passing through
the given point A, and cutting the given line at D; draw
D F, and produce it to cut the
circle at E; and draw A E the
perpendicular.

Note.—This second method
is useful when the point A is
at or near one end; and in prac-
tice, it is expedient in the first
place to strike out a preliminary
arc, of any convenient radius, from the point A, as any
point in that arc may be chosen for the centre F, with the
certainty that the arc from this centre will pass through
A, without the delay of adjusting the point of the compass
to it. This expedient is of general use where an arc is to
be passed through a given point, and particularly if the
point of the pencil be round or misshapen, and therefore
uncertain.

3 d Method.—From A describe an arc E C, and from E,
with the same radius, the arc A C,
cutting the other at C; through C draw
a line E C D, and set off C D equal to
C E, and through D draw the perpen-
dicular A D as required.

Note. — This method, like the pre-
vious one, is useful when the point A
is at one end.

4th Method.—From the given point A, set off a distance
A E equal to three parts, by any
scale ; and on the centres A and E,
with radii of four and five parts re-
spectively, describe arcs intersecting
at C. Draw A C for the perpendicular
required.

Note.—This method is most useful
on very large scales, where straight edges are inapplicable,
as in laying down perpendiculars on the ground. Beams,

(Fig. 65.)

/ /'
/ /

/

//

(Fig. 66.)

columns, and the like, may be set vertically and per-
pendicularly by the same method. The numbers 3, 4, 5,
are, it is to be observed, taken to measure respectively—-
the base, the perpendicular, and the slant side of the
triangle A E C. Any multiples of these numbers may
be used with equal propriety, as 6, 8, 10, or 9, 12, 15,
whether feet, yards, or any other measure of length.

Problem V.—To draw a perpendicular to a straight
line, from a point without it.

1st Method.—From the point A, with a sufficient radius,
(Fig. 67.) cut the given line at F and G;

and from these points describe
arcs cutting at E. Draw the
perpendicular A E.

Note.—If there be no room
below the line, the intersection
may be taken above; that is,
between the line and the given
point. This mode is not, however, likely to be as exact
in practice as the one given.

2d Method.—From any two points B, C, at some dis-
tance apart, in the given
line, and with radii B A,
C A, respectively, describe
arcs cutting at A D. Draw
the perpendicular A D.

Note.—This method is
most useful where the given
point is opposite the end
of the line, or nearly so. The second centre, C, may be
selected at or near G.

Problem VI.—To draw a straight line parallel to a
given line, at a given distance apart.

From the centres A, B, in the given line, with the
given distance as radius, de-
scribe arcs C D; and draw the
parallel line C D touching the
arcs.

A B Note.—The method of draw-

ing tangents will be afterwards shown ; meantime, in all
ordinary cases, the line C D may be drawn by simply
applying a straight edge by the eye.

Problem VII.—To draw a parallel through a given
point.

1st Method.—With a radius equal to the distance of

the given point C from the
given line A B, describe the
arc D from B, taken con-
siderably distant from C;

A B draw the parallel through

C to touch the arc D.

a/ (Ffy-71.)_2d Method.—From A, the

/ ! given point, describe the arc

/ / F D, cutting the given line

i / at F; from F, with the same

“e——----p-- radius, describe the arc E A;

and set off F D equal to E A. Draw the parallel through
the points A, D.